Inhomogeneous solutions in $f(T,B)$ gravity

Cosmology have been enhanced in the last decade from an enormous increase of new observational surveys whose methods for handling them are still a challenging endeavor, not only because the fundamental nature of the dark sector is still an open question, but also from the persisting tensions between the values measured/inferred of $H_{0}$ and $f\sigma_{8}$ and anomalies in early-time observations [1, 2].

On the local universe side, there is a wide variety of dark energy proposals to explain the cosmic accelerated expansion under the General Relativity (GR) framework while considering gauge invariant cosmological perturbations on a Friedmann-Lemaître-Robertson-Walker (FLRW) background. As it is standard, this acceleration behaviour can be obtained by assuming, for example, a positive cosmological constant $\Lambda$, extra dark fluids with negative pressure, scalar fields or modifications of GR, among others [3, 4]. In this latter line of thought, extensions of Teleparallel Gravity (TG) [5, 6] have also been considered to explain the cosmic acceleration, and furthermore, to even alleviate the persisting cosmological tensions, all this from the geometry of the theory itself, i.e. without evoking exotic fluids or $\Lambda$ [7]. Even more, TG offers several viable cosmological proposals that have show tight constraints using current observations [8, 9, 10].

TG is a theory of gravity [11] in which the curvature-based description of gravity, used in GR, is enhanced by torsion, which involves replacing the Levi-Civita connection with its Weitzenböck connection analog. The name of teleparallelism or distant parallelism comes from the geometrical description of vectors in the Weitzenböck space-time [12], where torsion can change the direction of such vectors [13]. Besides, TG is a gauge theory for the translations group [14], so it is possible to represent the gravitational field with a translational gauge potential that appears within the non-trivial part of the tetrad field, i.e. keeps torsion different from zero.

The noteworthy complexity of the Einstein equations and the ones from modified theories of gravity lead us to employ new numerical methods to solve them when evolving with the rest of the matter/energy content of the Universe. Analytical and semi-analytical solutions provide useful toy scenarios to obtain asymptotic descriptions of cosmic structures using $\Lambda$CDM as background.

The simplest exact solutions providing this dust-like description are the spherically symmetric Lemaître-Tolman-Bondi (LTB) models [15, 16], whose metric also admits non-trivial solutions with non-zero pressure [17]. These models were used to explain cosmic acceleration through the effect of large-scale inhomogeneity [18]. These models were so-called Big Void, since they assumed our location near the center of a giant 1 Gpc spherical void. These solutions have also been very useful as toy models to study structure formation [19, 20, 21, 22], as they can also be conceived as exact non-linear perturbations of a FLRW background, reducing in the linear regime to standard cosmological perturbations in the isochronous comoving gauge [17, 23].

According to the above ideas, in this paper we study the compatibility between TG theories and LTB models. While there is an extensive literature on LTB models in GR, only a few consider inhomogeneous solutions in TG theories, e.g to analyse the possibility to describe the dark energy dynamics without introducing a cosmological constant [5] and discuss the problem of energy-momentum localization in TG [24]. Our main interest is to examine how the known properties of LTB solutions can be affected by the torsion contribution acting as a force in TG theories. Along this work, we refer to quantities with/without a white circle those upon them are calculated with the Levi-Civita/Weitzenböck connection, respectively. The signature used is $(-,+,+,+)$. Also, we will consider geometric units ($c=G=1$).

The paper is organized as follows. In Sec. 2 we introduce the LTB models constructed from the Schwarzschild solution. In Sec. 3 we described the main principles of TG, the Weitzenböck gauge and a tetrad formalism which obeys this gauge and maintains the spherical symmetry. In Sec. 4 we discuss LTB-like solutions in TG for a couple of $f(T,B)$ models and their cosmological implications. Finally, in Sec. 5 we present concluding remarks.

The Schwarzschild metric in the standard static coordinates can be read as

where $M_{0}$ denotes the Schwarzchild mass. The radial timelike geodesics of this spacetime follows

where $\kappa_{0}$ is the conserved energy in each geodesic. A new coordinate system $(\tau,r)$ based on the world lines of observers following radial geodesics can be constructed from Eq. (2) [16]. Under this coordinate system the proper time $\tau$ is the time coordinate, while $r$ is a continuous variable that label each geodesic. As the binding energy $k$ is a continuous parameter that varies along the geodesic congruence in the $(\tau,r)$ plane, it can be shown that $\kappa=\kappa(r)$ and $Y=Y(\tau,r),\,\,t=t(\tau,r)$. The coordinate transformation $(t,Y)\to(\tau,r)$ obtained by Eq. (2) takes the Schwarzschild static metric into the following time dependent form:

where $\kappa=\kappa(r)$ and $Y^{\prime}\equiv\left[\partial Y/\partial r\right]_{\tau}$, with $Y(\tau,r)$ satisfying

where $\dot{Y}\equiv u^{\mu}Y_{,\mu}=\left[\partial Y/\partial\tau\right]_{r}$ [25] and $\tau_{B}(r)$ an integration constant characterizing the value of $\tau$ for which $Y=0$, i.e. the equivalent to the Big Bang singularity in FLRW models. In these coordinates notice that the Schwarzschild radius $Y_{s}=2M_{0}$ is a regular point.

The Schwarzschild mass, $M$, can be interpreted in spherical coordinates of Eq. (3) as an integrated distribution $M_{0}=\int_{V}{\rho\,d^{3}x}$ where $\rho=M_{0}\delta^{3}(0)/(r^{2}\sin\theta)$. By considering a continuous density instead of a distributional density (which is equivalent to considering $M_{0}\,\rightarrow\,M=M(r)$) and taking as the $4$-velocity $u^{\mu}=\left[dx^{\mu}/d\tau\right]_{r}=\delta^{\mu}_{\tau}$, for observers comoving along the geodesics. The substitution of Eqs. (3)–(4) in the Einstein equations, $G_{\mu\nu}=8\pi\Theta_{\mu\nu}$ yields

These relations characterize LTB models with a dust source. The LTB metric has a natural FLRW background, defined in the space of parameters by the election of the free functions in the following way [15]

where $a(\tau)$ is the FLRW scale factor in conformal time, $\Omega_{m0}=\Omega_{m}(\tau)_{\tau=\tau_{0}}$ is the matter density parameter evaluated at current time and $\tau_{0}$ is current time defined by $a(\tau_{0})=1$. $H_{0}=H_{\tau=\tau_{0}}=\dot{a}(\tau)/a(\tau)_{\tau=\tau_{0}}$ is the Hubble parameter and $k$ is the constant curvature of the space–like hypersurfaces orthogonal to $u^{a}$. In this way, Eq. (4) in the FLRW limit Eq. (6) reads

with $\Omega_{k,0}$ the spatial curvature density.

TG arises as a gauge theory of the group of translations based on Noether’s Theorem since the energy–momentum current is covariantly conserved if the Lagrangian is invariant under space-time translations [26, 11], and is a geometric theory of gravity where the gravitational field is mediated by the torsion and not by the curvature, so it requires that the components of the $2$-form curvature vanish identically. The dynamical field in this case is the tetrad, and is related to the gauge potential of the translations group [27]. The tetrad and the metric are related by the following expression

As we can notice, if we perform a local Lorentz transformation on the tetrad $e^{A}_{\enskip\mu}\to e^{B}_{\enskip\mu}\Lambda(\mathbf{x})_{B}^{\,A}$, the metric remains invariant under such transformations.

Since different tetrads related upon local Lorentz transformation give the same metric, TG has to be local Lorentz invariant [28]. This requirement gives as a consequence an extra purely inertial degree of freedom called the spin connection $\omega_{\enspace B\mu}^{A}$, which is a $1$-form that assumes values in the Lie Algebra of the Lorentz group [28]. In this context, TG consist of a geometrical setup that includes a manifold $M$, a tetrad $\mathbf{e}=\{e^{A}\}^{3}_{A=0}$ and a spin connection $\boldsymbol{\omega}=\{\omega^{A}_{\hskip 4.26773ptB}\}_{A,B=0}^{3}$ [6]. Once a coordinate chart $\{x^{\mu}\}$ is given on the spacetime manifold $M$ and a second one $\{x^{A}\}$ is given on the Minkowski space as a fiber bundle on $M$, we can write the tetrad and the spin connection, respectively as $e^{A}=e^{A}_{\enspace\mu}dx^{\mu}$ and $\omega^{A}_{\hskip 4.26773ptB}=\omega^{A}_{\hskip 4.26773ptB\mu}dx^{\mu}$ [28, 6]. In TG, the linear connection on $M$ is no longer the Levi-Civita connection but the Weitzenböck connection defined by

where $e_{\enspace\mu}^{A}$ is the tetrad field, $E_{A}^{\enspace\sigma}$ the transpose tetrad field defined by $E_{A}^{\enspace\mu}e_{\enspace\nu}^{A}=\delta^{\mu}_{\nu}$. Greek indices refer to space-time coordinates while capital Latin letters stand for tangent space indices. The condition that the components of the $2$-form curvature vanish identically can only be achieved by the purely inertial spin connection [29, 6]

The spin connection given by Eq. (10) can be chosen to satisfy the Weitzenböck gauge $\omega_{\enspace B\mu}^{A}=0$ if a local Lorentz transformation is performed, since it is a transformed zero spin connection in an arbitrary Lorentz frame [30] and then, the tetrad becomes the only fundamental field. This approach is known as the pure tetrad formalism [26, 11], which breaks the local Lorentz invariance of the theory and can led us to a sensible choice of the tetrad field on its extensions. This issue is so-called the choice of good and bad tetrads [31, 32, 33].

With the choice of tetrad Eq. (19) and considering a dust source $\Theta_{\mu\nu}=\rho u_{\mu}u_{\nu}$, with $\rho$ the matter–energy density, the diagonal components of Eq. (15) can be read as:

The only non-zero component of the antisymmetric part of the field equations is given by

From this latter notice that $\dot{f}_{T}=-\dot{f}_{B}$ and $f_{T}^{\prime}=-f_{B}^{\prime}$ are solutions, which imply that $f(T,B)=f(T-B)-T+B=f(R)+\textrm{TEGR}$. In order to have a non-trivial form of the functional we must solve Eq. (23) with a generic choice of $f(T,B)$, which will be the initial point for our following analysis.

In this paper we have studied the effects of torsion-based models, namely $f(T,B)$ gravity, on the spherically symmetric LTB dust models, which is a simple exact solution providing a dust description of cold dark matter.

We begin by discussing the LTB solution from Schwarzschild solution in GR, which provides a background consistency check to test the solutions in $f(T,B)$ theories and allows an ansatz for the resulting PDE’s. As part of this discussion, we briefly introduce the principles of TG in order to analyse the most general tetrad that recovers the LTB metric. In such analysis, the $\text{SO}(3)$ symmetry is Weitzenböck gauge-compatible as a consequence of a proper choice of parameters. Also, we showed that the $tr$ component of the antisymmetric part of the field equations does not vanish identically, and therefore the symmetric part of the field equations and the non-zero component of the antisymmetric part of the equations have to be solved simultaneously for particular choices of $f(T,B)$.

As we are interested in viable cosmological scenarios, we reduced our analysis to the spatially flat case and considered a couple of $f(T,B)$ functionals, e.g. modified versions of Power Law (29) and Mixed Power Law (40) models, which have proved to be successful in reproducing cosmological scenarios such as the late cosmic acceleration and solve the $H_{0}$ tension.

The motivations for studying inhomogeneous models in TG lies in the fact that we can construct toy models to analyse structure formation and the accelerated cosmic expansion in local regimes. To study the structure formation scenario, we proposed the ansatz (31) which is inspired by the standard Schwarzschild solution. In this case, we obtained that $l=4$ is a solution that recovers $f(T)$ gravity with an exact closed form of the radial function $Y$ and a continuous and regular equation for the mass-energy density in terms of the radial function $Y$ and the free parameters of the $f(T,B)$ functional. Nevertheless, this solution modifies the Friedmann equation and sets specific constraints to reproduce a viable structure formation according to the observations. In particular, the case $l\neq 4$ can be solved for $\alpha=0$, where the solution is FLRW with $l=1$. For the Mixed Power Law model, we consider $n+m=1$, $(l-4)n-2m=0$ and $n=2^{\frac{m}{2}}$, where only the first option provides a constant $l$ without imposing additional constraints on $Y$ or $M$.

Furthermore, we found two possibles solutions for $l=1$ and $l=4$, nevertheless, the case $l=4$ implies a vanishing boundary term and by consequence a vacuum solution. The case $l=1$ does not modify Hubble’s law and gives the standard GR solution for the radial function $Y$. It is remarkable to mention that all constraints contain $l=4$ as a solution and only $n+m=1$ includes the case $l=1$. Here, the mass-energy density takes the form of the mass-energy density of GR plus an extra term. Here, this extra term produces a mass defect in comparison to GR when $f_{0}>0$. As an extension of this analysis, each of our set of parameters can be constrained using current observations, study that will be reported elsewhere.

CE-R acknowledges the Royal Astronomical Society as FRAS 10147. CE-R, A.A and GAR-F are supported by PAPIIT-DGAPA UNAM Projects IA100220 and TA100122. SN, A.A and GAR-F acknowledge financial support from SEP–CONACYT postgraduate grants program. RAS acknowledges support from PAPIIT-DGAPA RR107015. This work is part of the Cosmostatistics National Group (CosmoNag) project. The Authors thank J. Levi Said, S. Bahamonde and A. Golovnev for enlightening discussions.